Global testing against sparse alternatives in time-frequency analysis

TL;DR

This paper introduces an over-sampled periodogram higher criticism test for detecting sparse periodic signals in complex time series, establishing its optimality and demonstrating its effectiveness through simulations.

Contribution

It proposes a new global testing method for sparse periodic effects, providing explicit detection boundaries and proving the test's asymptotic power and universal optimality.

Findings

The OPHC test is asymptotically powerful within the detection boundary.

Over-sampling by O(log N) achieves universal optimality.

Numerical simulations confirm the test's effectiveness.

Abstract

In this paper, an over-sampled periodogram higher criticism (OPHC) test is proposed for the global detection of sparse periodic effects in a complex-valued time series. An explicit minimax detection boundary is established between the rareness and weakness of the complex sinusoids hidden in the series. The OPHC test is shown to be asymptotically powerful in the detectable region. Numerical simulations illustrate and verify the effectiveness of the proposed test. Furthermore, the periodogram over-sampled by $O(\log N)$ is proven universally optimal in global testing for periodicities under a mild minimum separation condition.